Characteristic subgroup of solvable group

From Groupprops
Jump to: navigation, search
This article describes a property that arises as the conjunction of a subgroup property: characteristic subgroup with a group property imposed on the ambient group: solvable group
View a complete list of such conjunctions | View a complete list of conjunctions where the group property is imposed on the subgroup

Definition

A subgroup H in a group G is termed a characteristic subgroup of solvable group if G is a solvable group and H is a characteristic subgroup.

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
characteristic subgroup of nilpotent group the subgroup is characteristic and the whole group is a nilpotent group follows from nilpotent implies solvable solvable not implies nilpotent -- use the group as a subgroup of itself |FULL LIST, MORE INFO
characteristic subgroup of abelian group the subgroup is characteristic and the whole group is an abelian group (via nilpotent) (via nilpotent) -- use the group as a subgroup of itself |FULL LIST, MORE INFO

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
normal subgroup of solvable group characteristic implies normal normal not implies characteristic in any nontrivial subquasivariety of the quasivariety of groups plus solvability is quasivarietal |FULL LIST, MORE INFO
solvable characteristic subgroup the subgroup is characteristic and solvable as a group in its own right. solvability is subgroup-closed the trivial subgroup in any non-solvable group. |FULL LIST, MORE INFO
solvable normal subgroup (via solvable characteristic, also via normal subgroup of solvable group) (via solvable characteristic, also via normal subgroup of solvable group) |FULL LIST, MORE INFO